Extremal metrics on toric manifolds.
Extremal metrics on toric manifolds.

Simon K. Donaldson, Imperial College, London
Fine Hall 314
Calabi introduced the notion of an extremal metric, as an optimum, canonical, Kahler metric in a given Kahler class on a compact complex manifold. Wellknown conjectures relate the existence of such a metric to "stability" conditions of the complex structure. In the case of toric manifolds these questionsinvolving differential geometry, PDE and algebraic geometrycan be formulated in an elementary way, in terms of convex geometry on the corresponding polytope. Moreover, very recent work of Chen and Cheng leads to a general existence theorem in this setting. In the lecture we will survey these developments and explain some of the further questions that arise.